The internet of things (IoT) is an emerging archetype of technology for the guaranteed quality of services (QoS). The availability of the uninterrupted power supply (UPS) is one of the most challenging criteria in the successful implementation of the service system of IoT. In this paper, we consider a fault-tolerant power generation system of finite operating machines along with warm standby machine provisioning. The time-to-failure for each of the operating and standby machines are assumed to be exponentially distributed. The time-to-repair by the single service facility for the failed machine follows the arbitrary distribution. For modeling purpose, we have also incorporated realistic machining behaviors like imperfect coverage of the failure of machines, switching failure of standby machine, reboot delay, switch over delay, etc. For the evaluation of the explicit expression for steady-state probabilities of the system, the only required input is the Laplace-Stieltjes transform (LST) of the repair time distribution. The step-wise recursive procedure, illustrative examples, and numerical results have been presented for the following different type of repair time distribution: exponential ($ M $), $ n $-stage Erlang ($ Er_{n} $), deterministic ($ D $), uniform ($ U(a, b) $), $ n $-stage generalized Erlang ($ GE_n $) and hyperexponential ($ HE_n $). Concluding remarks and future scopes have also been included.

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\ begin {document} $ \ bf {M / G / 1} $ \ end {document} 不完美的容错加工系统

物联网（IoT）是一种新兴的技术原型，可确保服务质量（QoS）。在成功实施物联网服务系统中，不间断电源（UPS）的可用性是最具挑战性的标准之一。在本文中，我们考虑了有限运行机器的容错发电系统以及热备机配置。假定每台运行和待机机器的故障时间都是指数分布的。单一服务设施对故障机器的维修时间遵循任意分配。出于建模目的，我们还结合了实际的加工行为，例如对机器故障的覆盖范围不完美，备用机器的切换故障，重启延迟，切换延迟等。为了评估系统稳态概率的显式表达式，唯一需要的输入是维修时间分布的Laplace-Stieltjes变换（LST）。对于以下不同类型的维修时间分布，已给出了逐步递归过程，说明性示例和数值结果：指数（$ M $），$ n $级Erlang（$ Er_ {n} $），确定性（ $ D $），统一（$ U（a，b）$），$ n $阶段的广义Erlang（$ GE_n $）和超指数（$ HE_n $）。结束语和将来的范围也已包括在内。并针对以下不同类型的维修时间分布提供了数值结果：指数（$ M $），$ n $阶段Erlang（$ Er_ {n} $），确定性（$ D $），统一（$ U（ a，b）$），$ n $阶段的广义Erlang（$ GE_n $）和超指数（$ HE_n $）。结束语和将来的范围也已包括在内。并针对以下不同类型的维修时间分布提供了数值结果：指数（$ M $），$ n $阶段Erlang（$ Er_ {n} $），确定性（$ D $），统一（$ U（ a，b）$），$ n $阶段的广义Erlang（$ GE_n $）和超指数（$ HE_n $）。结束语和将来的范围也已包括在内。